We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs Kaufmann et al., 2024 can be used to count the crossings in terms of the number n of vertices. As a main result, we show that every 3-plane drawing has at most 5.5(n-2) crossings, which is tight. In particular, it follows that every 3-planar graph on n vertices has crossing number at most 5.5n, which improves upon a recent bound Bekos et al., 2024 of 6.6n. To apply the Density Formula, we carefully analyze the interplay between certain configurations of cells in a 3-plane drawing. As a by-product, we also obtain an alternative proof for the known statement that every 3-planar graph has at most 5.5(n-2) edges.
Goetze et al. (Wed,) studied this question.
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